Abstract
This paper describes the algorithms and theory behind a new code for vector Sturm-Liouville problems. A new spectral function is defined for vector Sturm-Liouville problems; this is an integer valued function of the eigenparameter λ which has discontinuities precisely at the eigenvalues. We describe numerical algorithms which may be used to compute the new spectral function, and its use as amiss-distance function in a new code which solves automatically a large class of regular and singular vector Sturm-Liouville problems. Vector Sturm-Liouville problems arise naturally in quantum mechanical applications. Usually they are singular. The advantages of the author's code lie in its ability to solve singular problems automatically, and in the fact that the user may specify the required eigenvalue by its index.
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References
F.V. Atkinson,Discrete and Continuous Boundary Problems (Academic Press, New York, 1964).
F.V. Atkinson, A.M. Krall, G.K. Leaf and A. Zettl, On the numerical computation of eigenvalues of Sturm-Liouville problems with matrix coefficients, preprint (1987).
M.H. Alexander and D.E. Manolopoulos, A stable linear reference potential algorithm for solution of the quantum close-coupled equations in molecular scattering theory, J. Chem. Phys. 86 (1987), 2044–2050.
N. Dunford and J.T. Schwartz,Linear Operators, Part II (Wiley Interscience, New York, 1988).
G.H. Golub and C.F. Van Loan,Matrix Computations (North Oxford Academic, Oxford, 1983).
P. Hartman,Ordinary Differential Equations (Johns Hopkins University Press, 1973).
L.Gr. Ixaru, The error analysis of the algebraic method for solving the Schrödinger equation, J. Comput. Phys. 9 (1972) 159–163.
L.Gr. Ixaru, M.I. Christu and M.S. Popa, Choosing step-sizes for perturbative methods of solving the Schrödinger equation, J. Comput. Phys. 36 (1980) 170–181.
T. Kato,Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften 32 (Springer, 1980).
L. Greenberg, A. Prüfer method for calculating eigenvalues of self-adjoint systems of ordinary differential equations, Parts 1 and 2, Technical Report, Department of Mathematics, University of Maryland (August 1991).
P. Hartman,Ordinary Differential Equations (Wiley, New York, 1964).
V. Hutson and J.S. Pym,Applications of Functional Analysis and Operator Theory (Academic Press, London, 1980).
R.D. Levine, M. Shapiro and B.R. Johnson, Transition probabilities in molecular collisions: Computational studies of rotational excitation, J. Chem. Phys. 52 (1970) 1755–1766.
J.V. Lill, T.G. Schmalz and J.C. Light, Imbedded matrix Green's functions in atomic and molecular scattering theory, J. Chem. Phys. 78 (1983) 4456–4463.
M. Marletta, Numerical tests of the SLEIGN software for Sturm-Liouville problems, ACM Trans. Math. Software 17 (1991) 481–490.
M. Marletta, Theory and implementation of algorithms for Sturm-Liouville computations, Ph.D. Thesis, Royal Military College of Science (1991).
M. Marletta and J.D. Pryce, Automatic solution of Sturm-Liouville problems using the Pruesss method, J. Comp. Appl. Math. 39 (1992) 57–78.
V.S. Melezhik, I.V. Puzynin, T.P. Puzynina and L. Somov, Numerical solution of a system of integrodifferential equations arising from the quantum-mechanical three-body problem with Coulomb interaction, J. Comput. Phys. 54 (1984) 221–236.
F. Mrugala and D. Secrest, The generalised log- derivative method for inelastic and reactive collisions, J. Chem. Phys. 78 (1983) 5954–5961.
L.I. Ponomarev, I.V. Puzynin, T.P. Puzynina and L. Somov, The scattering problem in quantum mechanics as an eigenvalue problem, Ann. Phys. 110 (1986) 274–286.
W.T. Reid,Ordinary Differential Equations (Wiley, New York, 1971).
J.D. Pryce, Error control of phase-function shooting methods for Sturm-Liouville problems, IMA J. Numer. Anal. 6 (1986) 103–123.
W.T. Reid,Sturmian Theory of Ordinary Differential Equations, Applied Mathematical Sciences 31 (Springer, 1980).
F. Rellich, Halbbeschränkte gewöhnliche Differentialoperatoren zweiter Ordnung, Math. Ann. 122 (1950/51) 343–368.
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Communicated by C. Brezinski